A331737 Multiplicative with a(p^e) = p^A000265(e), where A000265(x) gives the odd part of x.

1, 2, 3, 2, 5, 6, 7, 8, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 24, 5, 26, 27, 14, 29, 30, 31, 32, 33, 34, 35, 6, 37, 38, 39, 40, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 54, 55, 56, 57, 58, 59, 30, 61, 62, 21, 8, 65, 66, 67, 34, 69, 70, 71, 24, 73, 74, 15, 38, 77, 78, 79, 10, 3, 82, 83, 42, 85, 86, 87, 88, 89, 30

Offset: 1

Author

Antti Karttunen, Feb 02 2020

Keywords

• nonn
• mult

Comments

a(n) is the largest exponential divisor of n (cf. A322791) that is an exponentially odd number (A268335). - Amiram Eldar, Nov 17 2022

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Brahim Mittou, New properties of an arithmetic function, Mathematica Montisnigri, Vol LIII (2022).

Crossrefs

Cf. A000265, A268335, A322791, A331738.

Programs

Mathematica

f[p_, e_] := p^(e/2^IntegerExponent[e, 2]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 17 2022 *)

PARI

A000265(n) = (n>>valuation(n,2));
A331737(n) = { my(f = factor(n)); prod(k=1, #f~, f[k, 1]^A000265(f[k, 2])); };

Formula

a(n) = n / A331738(n).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((1-1/p) * Sum_{k>=1} p^(2^k - 1)/(p^(2^(k+1)-2) - 1)) = 0.3953728204... . - Amiram Eldar, Nov 17 2022